Exact Asymptotics for Learning Tree-Structured Graphical Models With Side Information: Noiseless and Noisy Samples

Tandon, Anshoo; Tan, Vincent Y. F.; Zhu, Shiyao

doi:10.1109/jsait.2020.3040552

Cited by 7 publications

(15 citation statements)

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“…The correlations between adjacent nodes ρ are set to 0.9, 0.7, and 0.5. Theorem 3 in [TTZ20] provides an exact asymptotic expression for the best possible error probability, and it serves as a baseline of the simulation; this is indicated as "Passive SCL: Theory" in Figs. 3, 4, and 5.…”

Section: Simulation Resultsmentioning

confidence: 99%

“…3, 4, and 5. Note that no other passive algorithm can perform better than that given by Theorem 3 in [TTZ20] (since this is based on the maximum likelihood or minimum error probability principle), so it also serves as a bona fide impossibility result for tree structure learning using passive strategies. deficiencies of the passive learning algorithm, especially when ρ is close to 1.…”

Section: Simulation Resultsmentioning

confidence: 99%

“…Thus, we only need to consider the path graph. Lemma 1 in [TTZ20] shows that for a three-node path 1 − 2 − 3 which is parameterized by P 1 (0) = 1 2 , P 2|1 (0|1) = P 2|1 (1|0) = θ and P 3|2 (0|1) = P 3|2 (1|0) θ1 , we have…”

Section: Additional Definition and Resultsmentioning

confidence: 99%

“…Remark 1. The error exponent for the passive learning algorithm on homogeneous Ising models does not depend on the tree structure T , but only depends on the edge correlation [TTZ20]. This is because the number of nodes is finite (and not growing) in our setting.…”

Section: Passive Structure Learning Algorithmmentioning

confidence: 95%

“…For homogeneous Ising trees, the passive error exponent K passive (T , ρ) can be expressed as an explicit function of ρ. Proposition 2 (Tandon, Tan, and Zhu [TTZ20]). For the homogeneous Ising tree with zero external field T parameterized by 0 < ρ < 1, the error exponent of the SCL algorithm can be expressed as…”

Section: Passive Structure Learning Algorithmmentioning

confidence: 99%

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Active-LATHE: An Active Learning Algorithm for Boosting the Error Exponent for Learning Homogeneous Ising Trees

Zhang¹,

Tandon²,

Tan³

2021

Preprint

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The Chow-Liu algorithm (IEEE Trans. Inform. Theory, 1968) has been a mainstay for the learning of tree-structured graphical models from i.i.d. sampled data vectors. Its theoretical properties have been well-studied and are well-understood. In this paper, we focus on the class of trees that are arguably even more fundamental, namely homogeneous trees in which each pair of nodes that forms an edge has the same correlation ρ. We ask whether we are able to further reduce the error probability of learning the structure of the homogeneous tree model when active learning or active sampling of nodes or variables is allowed. Our figure of merit is the error exponent, which quantifies the exponential rate of decay of the error probability with an increasing number of data samples. At first sight, an improvement in the error exponent seems impossible, as all the edges are statistically identical. We design and analyze an algorithm Active Learning Algorithm for Trees with Homogeneous Edges (Active-LATHE), which surprisingly boosts the error exponent by at least 40% when ρ is at least 0.8. For all other values of ρ, we also observe commensurate, but more modest, improvements in the error exponent. Our analysis hinges on judiciously exploiting the minute but detectable statistical variation of the samples to allocate more data to parts of the graph in which we are less confident of being correct.

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Section: Simulation Resultsmentioning

confidence: 99%

Section: Simulation Resultsmentioning

confidence: 99%

Section: Additional Definition and Resultsmentioning

confidence: 99%

Section: Passive Structure Learning Algorithmmentioning

confidence: 95%

Section: Passive Structure Learning Algorithmmentioning

confidence: 99%

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